Cracking tales of historical mathematics and its interplay with science, philosophy, and culture. Revisionist history is galore. Contrarian takes on received wisdom. Implications for teaching. Informed by current scholarship. By Dr. Viktor Blåsjö.
History of mathematics research with iconoclastic madcap twists
Reviel Netz’s New History of Greek Mathematics contains a number of factual errors, both mathematical and historical. Netz is dismissive of traditional scholarship in the field, but in some ways represents a step backwards with respect to that tradition. I argue against Netz’s dismissal of many anecdotal historical testimonies as fabrications, and his “ludic proof” theory.
A new book just appeared: A New History of Greek Mathematics, by Stanford Professor Reviel Netz, Cambridge University Press. Let’s do a book review.
It will be a critical review. The main theme will be the sciences versus the humanities. Note the title of the book: “a New History.” Netz’s “New History” represents the new humanities-centred dominance in the field. As opposed to the “old” histories written by more mathematically oriented people. In my opinion, “new” does not mean better in this case. And I will tell you why.
Let’s start by attacking a city. The enemy are hunkering down behind their city walls. We are going to have to scale the walls with ladders. How long should we make the ladders? The ancient historian Polybius has the answer:
“The method of discovering right length for ladders is as follows. … If the height of the wall be, let us say, ten of a given measure, the length of the ladders must be a good twelve. The distance from the wall at which the ladder is planted must, in order to suit the convenience of those mounting, be half the length of the ladder, for if they are placed farther off they are apt to break when crowded and if set up nearer to the perpendicular are very insecure for the scalers. … So here again it is evident that those who aim at success in military plans and surprises of towns must have studied geometry.”
Great stuff. But Netz gets it wrong, in my opinion. Here is how he concludes:
“And then, of course, we are supposed to apply – Polybius leaves this implicit – Pythagoras’s theorem.” (223)
I don’t think so. I don’t think that’s what Polybius intended.
Sure enough, you can solve for the length of the ladder using the Pythagorean Theorem, but that is a clumsy and inefficient way to do it. If you did this the modern way you would need to do some algebra followed by some calculation involving a square root. They didn’t have calculators on their phones back then, you know. Do you expect carpenters in the military to be able to calculate square roots by hand?
In fact, Polybius has already told you everything you need to know with his numerical example. If the wall is 10, the ladder should be 12, he says. But it scales! So what Polybius is really saying is that, whatever the height of the wall is, the ladder is always 20% longer than that. That’s all you need to know. No Pythagorean Theorem needed.
Those numbers are a rule of thumb. You can also do it more exactly if you want, according to Polybius’s more theoretical characterisation of the optimal length. But you don’t need the Pythagorean Theorem for that either. There’s a much better way, that you can easily teach to an illiterate carpenter in five minutes.
Draw an equilateral triangle, just as Euclid does in Proposition 1 of the Elements. Cut it down the middle. Now you have a right-angled triangle, where the base is exactly half of the hypothenuse. This corresponds precisely to Polybius’s rule: the distance along the ground is half the length of the ladder.
So now we have a scale model of what we want. The height down the middle of the equilateral triangle represents the city wall; the side of the equilateral triangle represents the ladder, and it is precisely half its own length from the foot of the wall, exactly as Polybius says it should be for optimal stability.
So if we are given that the height of the wall is for example 10 meters, then we divide the height of the triangle into ten equal parts. We take a blank ruler and mark those ten marks on it. Then we take this ruler, with this length unit, and measure the hypothenuse of the triangle. However many marks long it is, that’s how many meters our ladder needs to be.
Piece of cake. Easy to improvise in the field without any specialised knowledge or tools. While Netz is busy trying to teach his carpenters the algebra of quadratic expressions and how to extract square roots, I have already scaled his walls using my much quicker methods. That is what you get when you put humanities people in charge of mathematics.
So I wouldn’t trust Netz when it comes to mathematics, even when he says “of course,” as he does here.
Here is another example: Did you know that parabolas are pointier than hyperbolas? At least if we are to believe Professor Netz. This claim occurs in a discussion of Archimedes. Archimedes studied solids of revolution obtained by rotating a conic section around its axis. Here are Netz’s words:
“In the case of a parabola, this will be of a more pointed shape; in the case of the hyperbola, this may be more bowl-like.” (140)
This is BS. Parabolas are not “more pointed” than hyperbolas.
This is clear for example from the following fact: you can draw a hyperbola having any two given lines as asymptotes and passing through any given point. So in other words, you can draw a V, an arbitrarily pointy letter V, and then pick an arbitrary point inside that V, for instance a point super close to the vertex of the V. Then there is always a hyperbola that fits inside the V and that passes through the designated point. You can hardly get any pointier than that, now can you? Yet parabolas are nevertheless “more pointed”, somehow, Netz apparently believes.
By the way, this fact I just mentioned, about constructing a hyperbola within a given V (that is to say, with given asymptotes), that is Proposition 4 of Book II of the Conics of Apollonius.
Or is it? Here we have another interesting point. It seems that this proposition was actually not in the original version of the Conics. Because Eutocius, in late antiquity, needs this theorem at a certain point and he says he better prove it since it’s not in the Conics of Apollonius. But then in the text we have of the Conics, what we call Apollonius’s Conics today, this proposition clearly is there, with the exact same proof.
And in fact the standard text that we call Apollonius’s Conics today comes to us only through that very same author, Eutocius, who wrote a commentary on the Conics and also preserved the text at the same time. So it seems that Eutocius inserted this proposition into Apollonius’s original text, because he had noticed in other works that it was a useful thing to prove.
Netz describes this correctly, which is all the more reason why he should know that a hyperbola can be as pointy as you’d like, since this follows immediately from this proposition that he discusses at length.
But anyway, there is another kind of error here in Netz’s discussion of this. The point that this proposition of the Conics is an insertion by Eutocius — that insight, says Netz, is due to Wilbur Knorr, Netz’s predecessor as a classics professor at Stanford.
“No one noticed that prior to Knorr” (431-432), says Netz.
But that is not true. Wilbur Knorr was not the first to discover this. In fact, Knorr clearly says so in his own article, the very article cited by Netz, which Netz has evidently not read very carefully. Already in the 16th century, Commandino, in his Latin edition of the Conics, very clearly and explicitly made the exact same point as Knorr, using the exact same evidence and arguments. And this in turn was cited in a 19th-century German edition of the Conics, as Knorr himself says. So Knorr didn’t discovery anything except what people had already known for hundreds of years.
This is not such an innocent mistake. How are we supped to trust anything Netz says if he makes blatantly false statements that are clearly and unequivocally seen to be factually incorrect by simply glancing at the very article that Netz himself cites in support of his own claims?
But it’s even more problematic than that. Because it’s clearly not just a random mistake. It is an ideologically driven error. By saying that Stanford humanities professor Wilbur Knorr was the first to make this important scholarly discovery, Netz is obviously indirectly boosting the impression that his own claims are important and novel, since he too is a Stanford humanities professor.
Netz is not only saying that Wilbur Knorr was the first to discover this particular thing. He is implicitly saying that earlier generations of scholars missed important insights, and that only people like him — Stanford humanities professors — are true experts.
That is of course the point of the title of the book: A *New* History of Greek Mathematics. In the past everybody did it wrong, and we need people like Netz to finally do it right. There is indeed a lot of explicit posturing to this effect throughout the book.
Let’s look at another example of this. Let me read a passage where Netz is attacking Thomas Kuhn’s account of the history of astronomy. Thomas Kuhn wrote in the mid-20th century and he worked on the history of science even though his PhD was in physics. So that is exactly the kind of people Netz wants to denigrate. He wants to say that only specialised humanities professors, with their “new” histories, are actual experts in the field.
Here is what Netz says about Kuhn: “Like most nonspecialists, Kuhn supposed …” See? I told you. It’s not just that Kuhn was wrong. It is that Kuhn epitomises the kind of people (people with a PhD in physics, for example) who need to be eliminated from the field because they make so many hopelessly naive assumptions without even realising it. Anyway, let’s continue with the quote:
“Like most nonspecialists, Kuhn supposed that Aristotle was broadly canonical from the beginning and that although the ancients offered various astronomical variations, these had all to agree with the Aristotelian framework. … This is wrong. In fact, Aristotle was not canonized throughout most of antiquity; Greek philosophers were in continuous, ever-shifting debate; the very practices of astronomy went through several stages in antiquity before they became stabilized through the ultimate canonization of Ptolemy – and of Aristotle – in Late Antiquity.” (487)
Indeed, I agree with Netz that mathematicians and scientists would have ignored Aristotle. Netz says it very well:
“In the second century BCE itself, Aristotle was marginal even within philosophy, let alone to a scientist such as Hipparchus. It is quite likely that Hipparchus never even read Aristotle’s Physics.” (346) Reassessing ancient science in this light, “we come close to imagining a very Galilean Hipparchus” (347).
Yes, perfect, I agree. That is exactly what I have said before about ancient science as well. Go Team Netz on that one.
But what about poor Kuhn whom Netz uses as a punching bag? Was he really so stupid? No.
I went to my copy of Kuhn’s book on the Copernican Revolution to check Netz’s accusations, and here is what I found. Here is a quote from Kuhn’s book:
“The great Greek philosopher and scientist, Aristotle, whose immensely influential opinions *later* provided the starting point for most medieval and much Renaissance cosmological thought.” (Kuhn, Copernican Revolution, 78)
So Kuhn says exactly the opposite of what Netz accuses him of “supposing”. *Later* Aristotle provided the starting point of scientific thought. Not “from the beginning.” Later. Exactly as Netz himself argues.
Here is another quote from Kuhn’s book that says the same thing:
“Aristotle said a great many things which later philosophers and scientists did not have the least difficulty in rejecting. In the ancient world there were other schools of scientific and cosmological thought, apparently little influenced by Aristotelian opinion. Even in the late centuries of the Middle Ages, when Aristotle did become the dominant authority on scientific matters, learned men did not hesitate to make drastic changes in many isolated portions of his doctrine.” (Kuhn, Copernican Revolution, 83)
There is no way you can read this and say that “Kuhn supposed that Aristotle was broadly canonical from the beginning,” that is to say, from his own lifetime onwards. Kuhn clearly says the opposite.
Netz’s accusation is just slander. So it’s the same in both the Knorr case and the Kuhn case: Netz makes false assertions and then cites sources that clearly and explicitly say the exact opposite of what Netz alleges.
At least in these cases Netz bothered to provide references at all. More often he doesn’t even do that. He allows himself the licence to make assertions at will, which readers are supposed to accept on his authority alone. Consider for example the following rant about the alleged bias of some unnamed “past scholarship”:
“In past scholarship, this Babylonian achievement [in astronomy] was sometimes dismissed as ‘merely’ practical, the Babylonians unfavorably compared with the Greeks in that they did not produce a geometrical account of the sky, hence no physical model, so, unlike the Greeks, ‘not real science’. This is obviously an absurd special pleading, where one defines as scientific whatever it is that the Greeks do and then reprimands the non-Greeks for failing to be Greek. The Babylonian theory is in fact directly analogous to the Greek mathematical theory of music – whose scientific significance no one doubts.” (326)
Well, no wonder that we need a “new” history of Greek mathematics then, amirite? That darn “past scholarship,” you know, they couldn’t think straight back then because they were so biased in favour of the Greeks. Or why sugarcoat it, why not just come out and say it: They were all racist back then, weren’t they? Thank God we have proper humanities-trained experts like Netz at last to save us from all of that. “A New History of Greek Mathematics”. Basically code for: The First non-Racist History of Greek Mathematics.
Well, yes, the argument that Netz refutes is indeed idiotic. But what is this so-called “past scholarship” that allegedly made this idiotic and basically racist assertion that Babylonian astronomy is “not real science” because it’s not geometrical? Who ever said that? No one I ever heard of.
Maybe Thomas Heath? If Netz is the “new” history of Greek mathematics, then Heath’s famous book is obviously the old one, written more than a hundred years ago.
But no. I looked it up. Even old Heath explicitly uses the phrase “Babylonian science” with approval (History I, 8). Of course it was “science”. Perhaps Thales, in his travels, learned of “Babylonian science”, for example, Heath says (Aristarchus of Samos, 18), in exactly those words.
So who, then, is Netz arguing against, except straw men that he has made up to present himself as the anti-racist saviour? I don’t know.
But enough bickering about that. Let’s turn to a big issue of major interpretative importance.
According to Netz, “Thales and Pythagoras did no mathematics whatsoever” (17). According to Netz, earlier generations of scholars naively believed in such fairy tales because they blindly trusted a single source:
“My predecessor Heath and many historians – up until the last generation – gave credence to the view according to which Thales, and then Pythagoras, made lasting contributions to mathematics. This derives almost entirely from Proclus’s commentary, which, because of its overall sobriety, was taken seriously even for such obviously unfounded assertions.” (423)
First of all, it is not true that this “derives almost entirely from Proclus’s commentary.” It is disturbing that Netz makes this false and self-serving statement. Just read Heath, whom Netz names in this very rant. Read Heath’s chapter on Thales. Heath goes through the sources explicitly. There are several sources about Thales as a mathematician that predate Proclus. And several of those testimonies, as well as passages in Proclus, are explicitly attributed to various specific earlier authors.
So it is not the case that earlier generations of scholars uncritically and blindly relied “almost entirely” on a single biased source, as Netz dishonestly and falsely claims.
Let’s look at Thales and Pythagoras in turn. Let’s start with Thales.
I have spoken before about how the idea of Thales as the originator of formal geometry makes good sense. The way I told it was based on two theorems attributed to Thales.
The first theorem is that a diameter cuts a circle in half. I described how one can show that using a very neat proof by contradiction. The appeal, obviously, would not have been the theorem as such, but the realisation that that kind of thing can be established by a very elegant and satisfying type of reasoning, namely a rigorous argument based on paying careful attention to the definitions of concepts such as circle and diameter, and the remarkable power of proofs by contradiction for proving this kind of thing. That is exactly the same aesthetic that one finds on the first pages of almost any modern mathematics textbook in abstract algebra, for example: proofs of basic results driven by carefully formulated definitions and tidy proofs by contradiction. It makes sense that people would fall in love with this aesthetic that has stood the test of time, and it makes sense that it would have begun with a basic theorem such as that the diameter bisects a circle. Just as ancient sources suggest.
A second theorem attributed to Thales is that a triangle inscribed in a circle with the diameter as one of its sides must be a right triangle. It is natural to arrive at this insight by playing around with ruler and compass. And the aha-moment would then have been that one can prove such things. Make a rectangle, draw a diagonal, draw the circumscribing circle. Now you are in business. From playing with shapes, you have arrived at a proof of a universal truth. Pretty cool. It makes sense that the idea of proving geometrical theorems might have started with something like that, as some ancient sources suggest.
I told my own version of this story, but in broad outline something like that is a pretty standard and well-known point of view. But Netz acts as if he has never heard of any of that. He pretends that people who believe that Thales initiated geometry are simply blindly taking Proclus’s word for it without having thought it through at all. Netz says so explicitly. Here are his words:
“I suggest here that Hippocrates’s works were among the earliest pieces of Greek mathematics ever to be written.” (48)
Ok, so that’s Hippocrates, considerably later than Thales, famous for a very technical and detailed argument about the areas of lunes, a kind of shape composed from circles. This looks a lot more like a specialised piece of technical geometry from a quite mature geometrical tradition. It seems like a very odd and obscure place to start with geometry altogether. In reply to this, Netz says:
“This might seem surprising. Could mathematics emerge like that – springing forth from Zeus’s head? Would we not expect mathematics to emerge in a more rudimentary form? In fact, I think this is precisely how we should expect mathematics to emerge: from Zeus’s head, fully armed. What would be the alternative? … Of course the very first mathematical works in circulation would contain remarkable, surprising results. Why else would you even bother to circulate them? I suspect that the counterfactual is sometimes not sufficiently carefully thought through here. Just what would a more rudimentary piece of mathematics look like? Would it prove some truly elementary results, such as, say, the equality of the angles at the base of an isosceles triangle? Why would anyone care about such a treatise, proving such a result?” (48)
It is baffling that Netz allows himself to make this lazy argument, as if no one had ever though those things through. He states these rhetorical questions as if no one had ever thought of any of that. But of course people have thought about that and they have compelling answers to Netz’s questions.
I just told you what the alternative to Netz’s narrative is and why it would make sense. And I am not the first person to say this. But Netz is too lazy to engage with alternative views seriously, so instead he dishonestly says that no one has ever thought through any alternative to his view.
So that’s Thales. Netz rejects a plausible interpretation of the Thales testimonies in ancient sources by dishonestly mischaracterising as hopelessly naive any scholars who adhere to such views.
Now Pythagoras. “Heath … had three full chapters on the mathematics of Thales and Pythagoras!” (22), Netz says triumphantly, suggesting that this is proof that his “new” history of Greek mathematics is sorely needed.
Anyone who believes in Pythagorean mathematics is stupid, according to Netz, and for this he relies on a famous book by Burkert. Here is how Netz describes it:
“[Burkert’s book] Lore and Science in Early Pythagoreanism … was a more careful, professionalized classical philology, keen to understand the authors we read not as mere parrots, repeating their sources, but instead as thoughtful agents who shape and retell the evidence as suits their agenda. Pythagoras, under such a reading, crumbles to the ground: almost everything … comes to be seen as the making of later authors from Aristotle on. Never mind: the historians of mathematics went on as before.” (23)
We hear the ideological overtones here. Burkert is Netz’s kind people: he is hailed as “professionalized.” By contrast, “the historians of mathematics went on as before”. That is to say, the mathematically trained people working on history of mathematics were a bunch of fools who didn’t even realise what fools they were, and we would be much better off if “professionalized” experts such as Burkert and, presumably, Netz himself, would be given a monopoly on expertise status in the field.
I do not agree with this, neither in terms of content nor ideology.
Regarding Pythagorean mathematics, since Netz doesn’t go into any more depth, I will now analyse Burkert’s book itself, which Netz accepts as gospel truth. A book review within a book review! Here we go.
According to Burkert, “the apparently ancient reports of the importance of Pythagoras and his pupils in laying the foundations of mathematics crumble on touch”. Not that phrase: “the foundations of mathematics.” I am going to criticise Burkert, and I am going to say that Burkert makes a naive and anachronistic assumption about what “the foundations of mathematics” are. (For page references for the quotes from Burkert, see my Operationalism article.)
When Burkert speaks of “the foundations of mathematics,” he takes for granted the traditional view that a core pillar of Greek geometry was its Platonist detachment from the physical world. As Burkert says, “Greek geometry assumed its final form in the context of [Plato’s] Academy … after Plato had … fixed its position as a discipline of pure thought.”
Indeed, Burkert’s arguments against Pythagoras’s mathematical significance are really arguments that he did not advocate a proto-Platonist philosophy of mathematics. Burkert’s overall thesis is that “that which was later regarded as the philosophy of Pythagoras had its roots in the school of Plato.” And indeed he proves convincingly that there was a clear tendency to distort history in this way in Platonic sources that is not consistent with more reliable sources outside this tradition.
For example, Burkert shows that when Proclus mentions Pythagoras in his “catalogue of geometers,” and attributes to him “a nonmaterialistic procedure” in mathematics, this, unlike the rest of the catalogue of geometers, is not based on the highly credible Eudemus. Instead it is copied from Iamblichus, that is to say, from the biased Platonic tradition.
From this it does not follow, as Burkert tries to argue, that Eudemus did not mention Pythagoras as a geometer at all. It follows only that Eudemus in this place likely did not associate Pythagoras with proto-Platonic views. This is enough to give Proclus the motivation to supplement his account with phrases from Iamblichus, even if Eudemus had mentioned Pythagoras in the original.
Burkert also observes that “Aristotle [says] expressly of the Pythagoreans [that] ‘they apply their propositions to bodies’—bringing out the distinction, in this regard, between them and all genuine Platonists.” Eudemus and Aristotle are clearly much more credible than the much later, more biased, and less intellectually accomplished Iamblichus and Proclus.
Thus Burkert’s arguments that Pythagoras’s alleged proto-Platonist philosophy of geometry is a fabrication of biased sources are quite convincing. However, it does not follow from this that the Pythagoreans did not take a profound theoretical and foundational interest in geometry altogether.
Burkert conflates these two conclusions, because he sees no alternative path to theoretical mathematics than through Platonic-style abstraction and detachment from physical considerations. Burkert believes that early work on geometrical constructions “is still not doing mathematics for its own sake”; rather, the “discovery of pure theory” was a later “accomplishment,” in his words.
If you have followed what I have said in the past you know that I reject this. Burkert is naive to assume a dichotomy between constructions and “pure theory.” Constructions were not the opposite of theory, and hence the opposite of “the foundations of mathematics,” as Burkert erroneously assumes. On the contrary, constructivism *was* the foundations of mathematics.
Once we admit that possibility, there is every reason to think that earlier mathematicians, such as the Pythagoreans, could very well have made profound and foundationally sophisticated contributions, while at the same time rejecting Platonising tendencies in the philosophy of geometry.
Indeed, when going beyond his convincing case against Pythagoras the Platonist, to the more general case of trying to minimise the significance of Pythagoras and his followers in the history of geometry, Burkert find himself on the back foot. He is forced to try to explain away Aristotle’s compelling statement that “the so-called Pythagoreans were the first to take up mathematics; they advanced this study, and having been brought up in it they thought its principles were the principles of all things.”
Burkert’s thesis leaves him little choice but to dismiss the centrality of mathematics implied by this statement as “a psychological conjecture of Aristotle, which the historian is not obliged to accept.” That Proclus was wrong is plausible enough, but having to postulate that Aristotle was wrong comes at a considerably higher cost. And while Burkert was able to discredit Proclus’s mention of Pythagoras in the catalogue of geometers, he cannot deny that numerous attributions of mathematical discoveries to Pythagoreans made by Proclus are indeed based on Eudemus and hence credible, by Burkert’s own admission. Thus even Burkert must admit that “Pythagoreans made significant contributions to the development of Greek geometry.” Yet he hastens to add: “but the thesis of the Pythagorean foundation of Greek geometry cannot stand.”
Once again Burkert’s argument is based on tacitly assuming a monolithic conception of what “the foundations of Greek geometry” consisted in. The constructivist reading of Greek geometry problematises this assumption. It shows that one cannot simply take for granted that “the foundations of geometry” means what modern authors think it should mean. Constructivism offers an alternative vision, according to which much early Greek geometry may very well have been eminently foundational, but in a sense different from that commonly assumed by modern observers. This at the very least raises the possibility that early traditions such as that of the Pythagoreans may have been more foundationally significant than Burkert’s argument admits.
So much for Burkert, whose judgement Netz accepts unconditionally. Far from being an unequivocal triumph of “professionalized” expertise over previous naiveté, as Netz would have it, Burkert’s account is itself naive and by no means unquestionable.
So Netz is fond of dismissing what the ancient sources say. All the stories about Thales and Pythagoras, that’s just so much fiction. To be sure, the sources are highly imperfect and definitely contain a lot of misinformation. Nevertheless, it is surely better to try to save some meaning in these stories than to almost take it as a point of pride to dismiss as much of it as possible, as if the more sources you dismiss the more sophisticated a historian you are.
In fact, Netz continues in the same vein for later Greek geometry as well. “The stories [about Archimedes] probably are fabricated,” (128) we are told.
Stories such as Archimedes’s use of the principles of hydrostatics to detect a fake gold crown, because it did not have the right density properties. That is the “Eureka!” story.
“Biographers concoct anecdotes, based on the contents of the authors’ works. This is clearly the case here. The story of the crown is a clear echo of Archimedes’s study of solids immersed in liquids, On Floating Bodies.” (129)
Now, how would this work exactly? Let us “think through the counterfactual,” as Netz admonished others to do above.
Ok, so Archimedes wrote a sophisticated technical work on floating bodies. For some reason. Certainly not because of fake gold crowns and such things, because those are just “concocted anecdotes.” I guess Archimedes just woke up one day as said to himself: I think I will prove a bunch of theorems about hydrostatics, which nobody has done before, because I’m a mathematician and I just do things arbitrarily for no reason with no connection to the real world.
So he wrote a detailed, hyper-mathematical treatise on floating bodies. Theorem-proof, theorem-proof.
And then, maybe hundreds of years later or whatever, another guy told himself: Hey hey, I’m a writer! I’m going to write about the history of mathematics, but I won’t find out actual facts about the history of mathematics. Instead I’m going to pour over these extremely technical treatises that very few people can understand, and I’m going to master their content in great depth, to the point where I will be able to invent out of thin air real-world scenarios that involve realistic, sophisticated applications of the complicated technical results found in these treatises. And my goal in doing so is to concoct a one-paragraph anecdote about for example Archimedes making a discovery in the bath that made him run naked through the streets. Haha, what a funny image to imagine him running and screaming eureka like that. Totally worth all those probably hundreds of hours that I had to spend studying very complicated mathematics and then designing and working out my own research-level applied mathematics problem just so that I could make this little joke about Archimedes running from the bath.
Well, that’s apparently what happened if we are to believe Netz.
I very much doubt that story tellers were ever that good. The story about Archimedes and the crown is really very good scientifically. The connection with the technical details of Archiemdes’s treatise is the real deal. If this is a “fabricated” anecdote “concocted” by a biographer, as Netz says, then that biographer was not only a story teller but one of the leading scientists of their age.
Look, I teach calculus regularly, and I always try to get students to think about the physical meaning of mathematical notions and interpret results in the context of a real-world scenario. And I can tell you that that is an uphill battle to say the least.
I don’t think Netz teaches calculus so I think he underestimates how hard it is to make up stories that simultaneously make perfect scientific sense.
It is quite easy, on the other hand, to make up stories that do *not* make scientific sense. And that bring us to another one of Netz’s theories.
Netz has another book called “Ludic Proof”. Ludic as in play, playfulness. According to this theory, mathematicians borrowed stylistic approaches from poets. Poets had a fondness for cleverly constructing narratives that led to surprising twist reveals. Mathematicians shared the same aesthetic, according to Netz.
Netz, in all seriousness, proposes that this could be the main reason why Archimedes did calculus-style calculations of areas at all, and why he even turned to mathematical physics at all. The root cause is supposed to be not ordinary scientific or mathematical motivations, but Archimedes’s desire to do mathematics in the style of the poets: mathematics was “written, always, against the background of wider literary currents, emphasizing subtlety and surprise” (218).
According to Netz this is why Archimedes did calculus-style calculations of areas and volumes:
“Archimedes … picked up a particular technique, first offered by Eudoxus, because its subtlety … made a certain kind of surprise especially satisfying. Hence the infinitary methods.” (218)
And this is also what made Archimedes apply mathematics to physics:
“[Archimedes] saw the possibilities of applying geometry to a seemingly unrelated field – the study of centers of the weight in solids … – because there was a particular payoff of subtlety and surprise to be obtained by the bringing together of apparently irreconcilable, maximally distinct fields of study. This was rather like Callimachus’s poetry! Hence the mathematization of physics.” (218)
So there you go, calculus and mathematical physics are just side effects of mathematicians pursuing their true goal, which was to imitate the poets. That is some tin-foil-hat level of crackpottery, in my opinion.
It is one thing that Netz previously advanced his bizarre theory in a specialised monograph. Of course it must be possible for scholars to try out unconventional ideas. But to put this crazy stuff in a survey history with a straight face, as if this was objective information that any beginner in the field needs to learn, that is quite irresponsible, in my opinion. Certain chunks of this book are not an introduction to the history of Greek mathematics, but an introduction to the pet theories of Reviel Netz that no one but him believes.
Let’s look at some specific mathematical examples that are allegedly all about surprise, according to Netz.
For example, Archimedes found the area of one revolution of the Archimedean spiral. How do you think he’s going to prove this? Well, you have probably already seen how Archimedes found the area of a circle. Naturally readers of his more advanced treatise on spirals would already have read his more basic treatise on the circle.
Archimedes found the area of a circle by cutting it into wedges, as it were. Equal-angle pizza slices all the way around.
Naturally it makes a lot of sense to try the same idea for the spiral. The Archimedean spiral is like a circle but with a linearly growing radius. In polar coordinates, the radius r is proportional to the angle theta.
So when we apply the method we used for the circle to the spiral we get a bunch of equal-angle wedges that gradually get bigger and bigger. The radius grows linearly with the angle, so the radii of the wedges form an arithmetic progression. For every equal increment of the angle, the radii increase by the same amount, let’s say alpha. And the Archimedean spiral starts with radius zero, so the radii go: alpha, 2 alpha, 3 alpha, etc.
To get the area of the spiral we have to add up all the wedges. Obviously the areas scale like the square of the radii. Linear scaling of distances means square scaling of areas. So since the radii went alpha, 2 alpha, 3 alpha, the areas will be proportional to alpha^2, (2 alpha)^2, (3 alpha)^2, and so on.
So to get the area we have to add up a series of squares, the squares of numbers in an arithmetic progression. Indeed, Archimedes has a theorem that does exactly this. That is his Proposition 10.
Did you find any of this “surprising”? Hardly. It was a predictable extension of the idea used for the circle. And the trick of getting a complicated area or volume by an infinite series sum of simpler components is also a well established trick. Archimedes used the same trick for the area of a parabolic segment, for example, and Euclid used it too, for example for the volume of a tetrahedron. The sum of a geometric series was the key ingredient in those cases, and now for the spiral we need the same kind of theorem but for the squares of numbers in an arithmetic progression. Very predictable and business as usual for a Greek geometer.
But Netz doesn’t think so. According to Netz, the reader of Archimedes’s treatise is not supposed to have been able to see those things and instead they are supposed to have been baffled by the introduction of Proposition 10, that is to say, the sum of the series. They are not supposed to have been able to realise that this series is obviously the same kind of area calculation by series that had been well-known at least since Euclid, and that the particular terms of the series obviously correspond to the most natural way of cutting up the spiral area.
Here is what Netz says:
“Archimedes aims at surprise. The key point is that as proposition 10 is introduced, Archimedes makes all efforts to disguise its potential application. … The key observation – that the sectors in a circle behave as the series of squares on an arithmetical progression – is not asserted in advance. Instead, the application of proposition 10 is postponed and revealed only at the very last minute when, introduced in the middle of proposition 24, it finally makes sense of the argument. … Everything is designed for the sake of this denouement where, finally, the narrative of the treatise would make sense in a surprising turn. Ugly, misshapen proposition 10 is really about sectors in spirals: the duckling was a swan all along!” (149)
I think this is nonsense. I don’t think Archimedes’s readers would have been surprised at all by any of this.
Today we teach our mathematics students: when you read a theorem, before you look at the proof, take a few minutes to think about how you would prove it. Then when you read the proof you will understand it much better. You will know which parts are easy and obvious, because you have already thought of those yourself. And you will appreciate the difficult parts because you have realised when trying to prove it yourself that certain steps would have to involve some real work.
I bet Archimedes’s readers did the same. They get a treatise by Archimedes, a key result of which is the area of a spiral. Indeed, the treatise comes with a prefatory letter by Archimedes himself where he highlights the key results, so obviously you know where it’s heading. You don’t just start reading cold from A to Z.
And if you follow the elementary advice that we teach all our undergraduates, without which you will never get far in mathematics, to try to prove it yourself before reading the solution, then you will very quickly realise that the obvious approach is to cut the spiral area into wedges and sum the components, which will obviously lead to a series of squares of numbers in an arithmetic progression. So when you get to Archimedes’s Proposition 10 you will be far from surprised. On the contrary, you knew all along that he would have to do this sum.
Let’s look at another example of a so-called “ludic proof.” If you point a parabolic mirror at the sun, all the rays are reflected toward a single point, the focus of the parabola. Diocles proved this, and the “ludic” part is that he first proved some properties of tangents and normals of a parabola, and only then introduced a line parallel to the axis, which represent the rays of the sun. Surprise! It was about rays of the sun all along. Who could ever have guessed that saying something about the tangent first would be relevant to this! Except of course someone who has read the title of the treatise and has basic mathematical competence.
Here is how Netz describes it:
“[Diocles’s proof of the focal property of the parabola is] palpably Archimedean. The same emphasis on subtle surprise – down to the intentional delay in the construction of the parallel line, so that, throughout the argument, we do not yet see the relevance of any of it for the optics of rays of the sun.” (215)
So the “surprise” is that basic properties of the tangent of the parabola are relevant to the optics of rays of the sun. What a shocking reveal! Since the solar ray had not been drawn yet, there is no way we could have known this, according to Netz.
Once again, any mathematically competent person who looks at this problem for five seconds will realise that of course it is going to involve the tangent. The notion that mathematically competent readers would not have been able to see the relevance of theorems about tangents for the optics of rays of the sun is ridiculous. And yet that notion is the corner stone of Netz’s ludic proof interpretation of this episode.
There is another bit of nonsense here as well. Diocles talks about the tangent of a parabola, but Archimedes also talked about the tangent of a parabola. Aha! Therefore Diocles’s proof “is really a brilliant variation on an Archimedean theme” (215), in Netz’s words.
This is a way of thinking that perhaps makes sense in literary history. Poets and playwrights like to draw inspiration from earlier masterpieces and rework their themes in a new way. Netz tries to do the same thing for mathematics, but in my opinion the results are nonsensical.
What Netz is saying is like saying that if Person A gives a mathematical argument involving the derivative of a quadratic function, and then Person B gives a completely different argument that has nothing to do with the first one except that it too involves the derivative of a quadratic function, then Person B’s argument is a variation on Person A’s theme.
That’s rubbish. Of course tangents of parabolas show up regularly in mathematics. That doesn’t mean that anyone who talks about the tangent of a parabola is subtly reworking what earlier authors have done. That may be how literature works, but it is not how mathematics works.
So, in this case as in so many others, Netz’s “new history” is what you get when you look at Greek mathematics through eyes attuned to the humanities but not to mathematics. Indeed, Netz’s description of the mathematics is factually wrong as well. Archimedes and Diocles both state the tangent theorem in terms of “the intercept between tangent and ordinate” (215), according to Netz. No, that’s not right. It’s the intercept between the tangent and the axis. Not ordinate, axis.
But it is not my goal to catalogue all the mathematical errors in Netz’s book. If you take a humanities professor as your guide to mathematics then you have only yourself to blame anyway.